Robust Iterative Learning Hidden Quantum Markov Models

TL;DR

This work addresses robust learning for quantum sequential models by introducing Adversarially Corrupted HQMMs (AC-HQMMs) and the Robust Iterative Learning Algorithm (RILA). RILA combines a corruption-filter (RCR-EF), batched stochastic Kraus-operator updates, and a derivative-free optimization framework with L1 penalties to ensure physical validity and improved convergence under adversarial perturbations. Across HQMM and HMM benchmarks, RILA outperforms the prior ILA and the classical EM on quantum-generated data, while remaining competitive with EM on classical data and proving robust under corruption. The approach provides a principled pathway for reliable quantum sequential learning with practical implications for quantum process modeling and noisy real-world data, and the codebase is publicly available for replication.

Abstract

Hidden Quantum Markov Models (HQMMs) extend classical Hidden Markov Models to the quantum domain, offering a powerful probabilistic framework for modeling sequential data with quantum coherence. However, existing HQMM learning algorithms are highly sensitive to data corruption and lack mechanisms to ensure robustness under adversarial perturbations. In this work, we introduce the Adversarially Corrupted HQMM (AC-HQMM), which formalizes robustness analysis by allowing a controlled fraction of observation sequences to be adversarially corrupted. To learn AC-HQMMs, we propose the Robust Iterative Learning Algorithm (RILA), a derivative-free method that integrates a Remove Corrupted Rows by Entropy Filtering (RCR-EF) module with an iterative stochastic resampling procedure for physically valid Kraus operator updates. RILA incorporates L1-penalized likelihood objectives to enhance stability, resist overfitting, and remain effective under non-differentiable conditions. Across multiple HQMM and HMM benchmarks, RILA demonstrates superior convergence stability, corruption resilience, and preservation of physical validity compared to existing algorithms, establishing a principled and efficient approach for robust quantum sequential learning.

Figures (22)

• Figure 1: Full quantum circuit to implement HMM in equation \ref{['eqn:HMM_quantum']}
• Figure 2: Diagram illustrating the stochastic generator for the HQMM as defined in Equation (\ref{['eq:transfer_matrix']}). Source: Figure 5 in monras2010hidden.
• Figure 3: Comparison of HQMM learning performance using patternsearch and fmincon optimization methods on a dataset generated from the (2,4)-HQMM. The plot shows true training (red dashed) and validation (red solid) log-likelihoods, alongside RILA learned values for patternsearch (black dashed for training, solid for validation) and fmincon (blue dashed for training, solid for validation). Iteration log-likelihoods are depicted with black circles (patternsearch) and blue squares (fmincon) across four batches.
• Figure 4: Comparison of HQMM learning performance using RILA and ILA across four batches on a dataset generated from the (2,4)-HQMM. The plot shows true training (red dashed) and validation (red solid) log-likelihoods, alongside RILA learned values (blue dashed for training, solid for validation) and ILA learned values (green dashed for training, solid for validation). Iteration log-likelihoods are depicted with blue squares (RILA) and green triangles (ILA) across four batches.
• Figure 5: Grouped bar plot comparing the performance of RILA (4 and 8 batches), ILA (4 and 8 batches), and EM on a dataset generated from the (2,4)-HQMM. Each bar represents the mean log-likelihood, with training (dashed-pattern, lighter bars) and validation (solid, darker bars) values shown side by side for each method. Error bars indicate standard deviations. True training (red dashed) and validation (red solid) log-likelihoods are overlaid as horizontal reference lines.

Theorems & Definitions (3)

• Definition 1: monras2010hidden
• Proposition 2
• proof